Biomedical Engineering Reference
In-Depth Information
is justified on physical grounds, meaning that the transformations include
only those physically possible between the two spaces. For example, if two
images of the same head are acquired, then for multimodality registration,
where the desired accuracy is of the order of one mm, only a rigid transforma-
tion is justified (unless there is intervening surgical resection).* For nonrigid
anatomy or intrapatient registration, the “registering” transformation may
not reflect the physical transformation in regions where no points are avail-
able. It is helpful to view the problem of underestimation in terms of a null
space. Regardless of the set of transformations, for sufficiently small errors
FRE can be expected to be an approximately linear function of the
fiducial
localization errors. The input space of FRE, as function of the FLEs, will
always include a null subspace, which is the space of patterns of FLEs that
can be completely compensated by one of the transformations in the set. If
the localization error pattern has a component in that null space, then FRE
will underestimate TRE.
The second problem is overestimation. This situation is common when random
components to FLE are uncorrelated among the
N
fiducials used in the regis-
tration. In that case, the influence on the transformation by the localization
error of some fiducials will tend to be cancelled by the influence of others.
Here again rigid transformations provide a simple example. Suppose that in
one image space FLE for half of the fiducials is 3 mm in the
N
x
direction, while
for the other half it is 3 mm in the
direction. The optimal registration
transformation will be the identity. For that transformation FRE will be 3 mm,
while the resultant TRE will be zero at all points. In more realistic situations
the cancellation is less complete.
The relationship between the expected FLE and the expected TRE depends
on both the number and placement of the fiducial points. For uncorrelated
FLEs, the relationship for rigid-body registration has been subject to conjec-
ture for many years but is now well understood in the RMS sense for isotro-
pic error patterns.
x
4,5
The relationship between FRE and FLE, which is due to
6
Sibson,
is surprisingly simple,
1/2
RMS FLE
RMS FRE
(
)
(
12
N
)
(
)
,
(6.3)
where
is again the number of fiducial points used in the registration. This
equation is of great importance because it provides a means to bridge the gap
from self-consistency to accuracy. (See also Section 6.3.2 for another means.)
Here FRE represents self-consistency, and TRE, whose statistics can be deter-
mined from those of FLE, represents accuracy. Equation 6.3 makes it possible
to validate the accuracy of the system without resorting to comparison with
any other system. This method for “bootstrapping” its accuracy gives the
point-based, rigid-body system a special place in the arena of validation.
A caveat should be given here regarding the problem of the null space of FRE
mentioned above. To the extent that there is a consistent, rigid displacement of
N
3
* The brain does pulsate within the closed skull by about 0.5 mm with the cardiac cycle.
